Question

In an atom two electrons move around the nucleus in circular orbits of radii \[R\] and \[4R\] . The ratio of the time taken by them to complete one revolution is:
(A) \[1:4\]
(B) \[4:1\]
(C) \[1:8\]
(D) \[8:7\]

Answer 1

Hint: We are asked to find the ratio of time taken by the two given electrons to move around the nucleus. We use the relationship between time period and radius of the orbit to find the ratio of the values. Time period of an electron around an orbit is defined as the time taken for the electron to complete one revolution.
Formulas used: The relationship between time period of an electron in an orbit is given as,
\[T \propto {\left( R \right)^{\dfrac{3}{2}}}\]
Where \[R\] is the radius of the orbit

Complete answer:
We can start by writing down the data given in the question.
The value of radii of the two orbits are given as, \[R\] and \[4R\]
We are asked to find the time taken by the electrons to completely move around the nucleus once. This is known as the time period of the electron.
Now we use the relationship between time period of an electron and the radius of the orbit, which is, \[T \propto {\left( R \right)^{\dfrac{3}{2}}}\]
In order to change the proportionality sign, we take two values of time (as asked in the question) and divide the both.
That would be like, \[\dfrac{{{T_1}}}{{{T_2}}} = {\left( {\dfrac{{{R_1}}}{{{R_2}}}} \right)^{\dfrac{3}{2}}}\]
Now we assign the values of radii to the question in the asked order \[\left( {R:4R} \right)\]
We take \[{R_1} = R\] and \[{R_2} = 4R\]
Substituting these values on the above equation, we have \[\dfrac{{{T_1}}}{{{T_2}}} = {\left( {\dfrac{{{R_1}}}{{{R_2}}}} \right)^{\dfrac{3}{2}}} \Rightarrow {\left( {\dfrac{R}{{4R}}} \right)^{\dfrac{3}{2}}} = {\left( {\dfrac{1}{4}} \right)^{\dfrac{3}{2}}} = \dfrac{1}{8}\]
In conclusion, the right answer is, option (C) \[1:8\]

Note:
In the Bohr model of the atom, electrons travel in defined circular orbits around the nucleus. The orbits are labelled by an integer, the quantum number n. Electrons can jump from one orbit to another by emitting or absorbing energy. The nucleus is considered to be positively charged as the particles present are positively charged protons and neutral charged neutrons.